Diagonal of Hexadecagon across Four Sides given Inradius Calculator

Formula Used:

\[ Diagonal_{across\ Four\ Sides} = \frac{\sqrt{2}}{\sin(\pi/16)} \times \frac{Inradius}{1 + \sqrt{2} + \sqrt{2 \times (2 + \sqrt{2})}} \]

Diagonal across Four Sides:

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1. What is Diagonal across Four Sides of Hexadecagon?

The diagonal across four sides of a hexadecagon (16-sided polygon) is the straight line joining two non-adjacent vertices that are separated by four sides. This measurement is important in geometric calculations and architectural design involving regular polygons.

2. How Does the Calculator Work?

The calculator uses the mathematical formula:

\[ Diagonal = \frac{\sqrt{2}}{\sin(\pi/16)} \times \frac{Inradius}{1 + \sqrt{2} + \sqrt{2 \times (2 + \sqrt{2})}} \]

Where:

\( \sqrt{2} \) — Square root of 2 (approximately 1.4142)
\( \sin(\pi/16) \) — Sine of π/16 radians
\( Inradius \) — Radius of the inscribed circle

3. Formula Explanation

Details: This formula derives from the geometric properties of regular hexadecagons. The trigonometric functions account for the angular relationships between sides and diagonals, while the square root terms represent the proportional relationships inherent in the polygon's structure.

4. Using the Calculator

Tips: Enter the inradius (distance from center to any side) of the hexadecagon in meters. The value must be positive. The calculator will compute the diagonal length across four sides.

5. Frequently Asked Questions (FAQ)

Q1: What is a hexadecagon?
A: A hexadecagon is a polygon with 16 sides and 16 angles. When regular, all sides and angles are equal.

Q2: What is the inradius of a polygon?
A: The inradius is the radius of the inscribed circle that touches all sides of the polygon from the inside.

Q3: How accurate is this calculation?
A: The calculation is mathematically exact for regular hexadecagons, with precision limited only by computational floating-point arithmetic.

Q4: Can this formula be used for irregular hexadecagons?
A: No, this formula applies only to regular hexadecagons where all sides and angles are equal.

Q5: What are practical applications of this calculation?
A: This calculation is useful in architecture, engineering design, computer graphics, and any field requiring precise geometric measurements of 16-sided polygons.